A159287 Expansion of x^2/(1-x^2-2*x^3).

0, 0, 1, 0, 1, 2, 1, 4, 5, 6, 13, 16, 25, 42, 57, 92, 141, 206, 325, 488, 737, 1138, 1713, 2612, 3989, 6038, 9213, 14016, 21289, 32442, 49321, 75020, 114205, 173662, 264245, 402072, 611569, 930562, 1415713, 2153700, 3276837, 4985126, 7584237, 11538800

Offset: 0

Author

Creighton Dement, Apr 08 2009

Keywords

• easy
• nonn

Comments

A floretion-generated sequence: 'i + 0.5('ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj').

From Greg Dresden, Nov 15 2024: (Start)

a(n) is the number of ways to tile a 2 X (n+1) board with L-shaped trominos and S-shaped quadrominos, where the first tile must be an upright L. For example, here are the a(7)=4 ways to tile a 2 X 8 board:

._____________. ._____________.

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|_|_||__|___| |_|___|||___|

._____________. ._____________.

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|_|_|_|___|| |__|___||__|_| (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Creighton Dement, Online Floretion Multiplier.
• Yüksel Soykan, A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences (2020) Vol. 4, No. 2, 227-251.
• Index entries for linear recurrences with constant coefficients, signature (0,1,2).

Crossrefs

Cf. A159284, A159285, A159286, A159288.

Essentially the same as A052947.

Programs

Magma

I:=[0,0,1]; [n le 3 select I[n] else Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 27 2018

Mathematica

LinearRecurrence[{0, 1, 2}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, May 24 2011 *)
CoefficientList[Series[x^2/(1-x^2-2x^3),{x,0,50}],x] (* Harvey P. Dale, May 29 2021 *)

PARI

a(n)=([0,1,0; 0,0,1; 2,1,0]^n*[0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

G.f.: x^2/(1-x^2-2*x^3).

a(n) = A052947(n-2). - R. J. Mathar, Nov 10 2009

a(n) = a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, May 23 2023

From Greg Dresden, Nov 17 2024: (Start)

a(2*n+1) = 2*a(n)^2 + 2*a(n+1)*a(n+2).

a(3*n+1) = Sum_{i=1..n} a(3*i-2)*2^(n-i). (End)